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In my lecture notes on quantum information processing my lecturer gives an example of composite systems as $|\phi\rangle=|0\rangle |0\rangle=|00\rangle$. I understand that if we have two qubits then its product state will be in 2n dimensional Hilbert space and I understand the 2 qubit state $|00\rangle$ to be represented in matrix representation as $\begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix}$ (if that is wrong please do correct my misunderstanding though). My question is about the notation $|0\rangle|0\rangle=|00\rangle$, how can we calculate this with matrices on the left-hand side we have a 2 by 1 matrix multiplied by a 2 by 1 matrix which cannot be calculated. I thought perhaps it was a matter of direct products but my calculation led to an incorrect result there too.

Could anyone clarify this for me, please?

Edit: It occurred to me that I think I'm mistaken about the matrix representation of $|00\rangle$, I think it would make more sense to be $\begin{pmatrix} 1 \\ 0\\0\\0 \end{pmatrix}$ in which case the direct product does work and I should take the notation $|0\rangle|0\rangle$ to be a shorthand for the direct product not the multiplication of two matrices, is that correct?

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$|0\rangle|0\rangle$ is actually a shorthand for $|0\rangle \otimes |0\rangle$ or $\begin{bmatrix} 1 \\ 0 \end{bmatrix} \otimes \begin{bmatrix} 1 \\ 0\end{bmatrix} $ where $\otimes$ stands for the tensor product or essentially the Kronecker product. To quote Wikipedia:

In mathematics, the Kronecker product, denoted by $\otimes$, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is denoted by the same symbol) from vectors to matrices, and gives the matrix of the tensor product with respect to a standard choice of basis. The Kronecker product should not be confused with the usual matrix multiplication, which is an entirely different operation.

Now the standard choice of basis for a two-qubit system is:

$\{|00\rangle = \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}, |01\rangle = \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix}, |10\rangle = \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix}, |11\rangle = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}\}$

If you wish, you can also represent the basis as (if you strictly take $\otimes$ as the outer product):

$\{|00\rangle = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, |01\rangle = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}, |10\rangle = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}, |11\rangle = \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}\}$

but then while carrying out calculations like determining the action of a quantum gate on a composite state you'd have to write the state using the vector representation (carefully read the linked Mathematics SE answer).

The key point here is that don't be bent on thinking of these linear algebraic operations in terms of matrices, but rather think in terms of linear maps. You'll get more comfortable with these things once you learn about tensors!

P.S: Kronecker product and outer product confusion

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